491:
457:
322:
593:
559:
285:
661:
627:
353:
521:
251:
484:
450:
423:
315:
360:
416:
278:
244:
668:
634:
600:
566:
528:
676:
This table can be found as follows. Geometric arguments show that the intersection matrix of the components of the fiber must be negative semidefinite, connected, symmetric, and have no diagonal entries equal to −1 (by minimality). Such a matrix must be 0 or a multiple of the Cartan matrix of an
1919:, the smooth locus of each fiber has a group structure. For singular fibers, this group structure on the smooth locus is described in the table, assuming for convenience that the base field is the complex numbers. (For a singular fiber with intersection matrix given by an affine Dynkin diagram
152:
Most of the fibers of an elliptic fibration are (non-singular) elliptic curves. The remaining fibers are called singular fibers: there are a finite number of them, and each one consists of a union of rational curves, possibly with singularities or non-zero multiplicities (so the fibers may be
2130:
164:
elliptic fibration. ("Minimal" means roughly one that cannot be factored through a "smaller" one; precisely, the singular fibers should contain no smooth rational curves with self-intersection number −1.) It gives:
1448:
839:
1631:
1294:
1178:
1851:
1741:
1065:
971:
2429:
906:
1521:
1370:
1783:
1673:
1559:
1220:
1107:
1896:
at such a fiber. That is, after a ramified finite covering of the base curve, the singular fiber can be replaced by a smooth elliptic curve. Which smooth curve appears is described by the
2674:
94:
Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and
2742:
2203:
1946:
2524:
2809:
2841:
1877:
1318:
997:
2769:
1966:
1469:
860:
2889:. It can be reversed, so fibers of high multiplicity can all be turned into fibers of multiplicity 1, and this can be used to eliminate all multiple fibers.
2014:
2892:
Logarithmic transformations can be quite violent: they can change the
Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces.
2591:
has nonnegative degree, and it has degree zero if and only if the elliptic surface is isotrivial, meaning that all the smooth fibers are isomorphic.
2744:. This is an explicit rational number between 0 and 1, depending on the type of singular fiber. Explicitly, the lct is 1 for a smooth fiber or type
1908:-invariant 1728 is the unique elliptic curve with automorphism group of order 4. (All other elliptic curves have automorphism group of order 2.)
161:
3415:
3354:
1391:
2236:
1948:, the group of components of the smooth locus is isomorphic to the center of the simply connected simple Lie group with Dynkin diagram
791:
1577:
1240:
1124:
1800:
1690:
1014:
920:
751:) changes as we go around a singular fiber. Representatives for these conjugacy classes associated to singular fibers are given by:
2340:
866:
3174:
1475:
1324:
3081:
1985:
1750:
1640:
1526:
1187:
1074:
3216:
2244:
2600:
2863:
703:, there are 2 components with intersection multiplicity 2. They can meet either in 2 points with order 1 (type I
714:, there are 3 components each meeting the other two. They can meet either in pairs at 3 distinct points (type I
2694:
2142:
3405:
1969:
3410:
1922:
2938:
is an elliptic fibration. We will show how to replace the fiber over 0 with a fiber of multiplicity 2.
61:
3400:
2473:
2778:
1893:
3330:
2814:
1860:
1301:
980:
114:
The product of any elliptic curve with any curve is an elliptic surface (with no singular fibers).
3322:
2851:
1904:-invariant 0 is the unique elliptic curve with automorphism group of order 6, and the curve with
95:
3105:
Barth et al. (2004), section V.10, Tables 5 and 6; Cossec and
Dolgachev (1989), Corollary 5.2.3.
1912:
744:
141:
3086:
2747:
1951:
1454:
845:
3375:
3338:
3298:
3258:
3226:
3184:
1973:
190:
3383:
3306:
3266:
153:
non-reduced schemes). Kodaira and Néron independently classified the possible fibers, and
8:
2563:
2321:, Takao Fujita (1986) gave a useful variant of the canonical bundle formula, showing how
154:
45:
33:
678:
1976:
of an elliptic fibration (the group of sections), in particular its torsion subgroup.
3349:
3314:
3212:
3170:
157:
can be used to find the type of the fibers of an elliptic curve over a number field.
136:
118:
75:, depending on the context). The fibers that are not elliptic curves are called the
3379:
3363:
3345:
3326:
3302:
3286:
3274:
3262:
3246:
3234:
3204:
3162:
2881:) of an elliptic surface or fibration turns a fiber of multiplicity 1 over a point
2843:, 5/6 for II, 3/4 for III, 2/3 for IV, 1/3 for IV*, 1/4 for III*, and 1/6 for II*.
2125:{\displaystyle K_{X}=f^{*}(L)\otimes O_{S}{\big (}\sum _{i}(m_{i}-1)D_{i}{\big )}.}
1989:
172:
125:
80:
68:
1972:.) Knowing the group structure of the singular fibers is useful for computing the
3371:
3334:
3294:
3254:
3222:
3180:
3158:
2847:
732:
688:
If the intersection matrix is 0 the fiber can be either an elliptic curve (type I
178:
The number of irreducible components of the fiber (all rational except for type I
131:
37:
29:
3192:
3029:). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibration
194:
72:
53:
49:
3208:
3166:
3394:
84:
57:
41:
3200:
2571:
102:
101:. They are similar to (have analogies with, that is), elliptic curves over
3350:"Modèles minimaux des variétés abéliennes sur les corps locaux et globaux"
3150:
2539:
1897:
740:
684:
The intersection matrix determines the fiber type with three exceptions:
186:
17:
2846:
The canonical bundle formula (in Fujita's form) has been generalized by
3367:
3146:
2318:
98:
2862:"Logarithmic transformation" redirects here. Not to be confused with
728:
490:
456:
321:
200:
The multiplicities of each fiber are indicated in the Dynkin diagram.
185:
The intersection matrix of the components. This is either a 1×1
25:
3290:
3277:(1964). "On the structure of compact complex analytic surfaces. I".
3250:
1443:{\displaystyle {\begin{pmatrix}-1&-\nu \\0&-1\end{pmatrix}}}
660:
626:
592:
558:
520:
483:
449:
422:
415:
352:
314:
284:
277:
250:
243:
128:
is elliptic, and has an elliptic fibration over the projective line.
3064:
is obtained by applying a logarithmic transformation of order 2 to
667:
633:
599:
565:
527:
359:
88:
2330:
depends on the variation of the smooth fibers. Namely, there is a
834:{\displaystyle {\begin{pmatrix}1&\nu \\0&1\end{pmatrix}}}
2223:
whose coefficients have greatest common divisor equal to 1, and
1626:{\displaystyle {\begin{pmatrix}-1&-1\\1&0\end{pmatrix}}}
1289:{\displaystyle {\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}}
1173:{\displaystyle {\begin{pmatrix}0&1\\-1&-1\end{pmatrix}}}
3052:
has a fiber of multiplicity 2 over 0, and otherwise looks like
1846:{\displaystyle {\begin{pmatrix}0&-1\\1&1\end{pmatrix}}}
1736:{\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}
1060:{\displaystyle {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}
966:{\displaystyle {\begin{pmatrix}1&1\\-1&0\end{pmatrix}}}
67:
The surface and the base curve are assumed to be non-singular (
56:, perhaps without a chosen origin.) This is equivalent to the
3317:(2007), "Kodaira's canonical bundle formula and adjunction",
2424:{\displaystyle K_{X}\sim _{\bf {Q}}f^{*}(K_{S}+B_{S}+M_{S}),}
2594:
The discriminant divisor in Fujita's formula is defined by
3145:
2004:. Over the complex numbers, Kodaira proved the following
1892:). This reflects the fact that an elliptic fibration has
1888:, IV, III, or II, the monodromy has finite order in SL(2,
901:{\displaystyle \mathbf {Z} /\nu \times \mathbf {C} ^{*}}
1900:
in the table. Over the complex numbers, the curve with
1516:{\displaystyle (\mathbf {Z} /2)^{2}\times \mathbf {C} }
1365:{\displaystyle (\mathbf {Z} /2)^{2}\times \mathbf {C} }
1809:
1699:
1586:
1400:
1249:
1133:
1023:
929:
800:
2817:
2781:
2750:
2697:
2603:
2476:
2343:
2145:
2017:
1954:
1925:
1863:
1803:
1753:
1693:
1643:
1580:
1529:
1478:
1457:
1394:
1327:
1304:
1243:
1190:
1127:
1077:
1017:
983:
923:
869:
848:
794:
147:
83:. Both elliptic and singular fibers are important in
60:
being a smooth curve of genus one. This follows from
3153:; Peters, Chris A.M.; Van de Ven, Antonius (2004) ,
2235:is projective (or equivalently, compact), then the
160:The following table lists the possible fibers of a
48:1. (Over an algebraically closed field such as the
2835:
2803:
2763:
2736:
2668:
2518:
2423:
2197:
2124:
1960:
1940:
1871:
1845:
1777:
1735:
1667:
1625:
1553:
1515:
1463:
1442:
1364:
1312:
1288:
1214:
1172:
1101:
1059:
991:
965:
900:
854:
833:
2306:; it is essential here that the elliptic surface
1984:To understand how elliptic surfaces fit into the
1778:{\displaystyle \mathbf {Z} /2\times \mathbf {C} }
1668:{\displaystyle \mathbf {Z} /3\times \mathbf {C} }
1554:{\displaystyle \mathbf {Z} /4\times \mathbf {C} }
1215:{\displaystyle \mathbf {Z} /3\times \mathbf {C} }
1102:{\displaystyle \mathbf {Z} /2\times \mathbf {C} }
3392:
2562:-divisors, using the identification between the
747:group of a smooth fiber (which is isomorphic to
3190:
2885:of the base space into a fiber of multiplicity
2857:
3033:is certainly not isomorphic to the fibration
2298:-linearly equivalent to the pullback of some
2285:). The canonical bundle formula implies that
2114:
2065:
743:1. The monodromy describes the way the first
731:around each singular fiber is a well-defined
3237:(1963). "On compact analytic surfaces. II".
2669:{\displaystyle B_{S}=\sum _{p\in S}(1-c(p))}
1979:
718:), or all meet at the same point (type IV).
707:), or at one point with order 2 (type III).
1884:For singular fibers of type II, III, IV, I
3331:10.1093/acprof:oso/9780198570615.003.0008
3114:Barth et al. (2004), Corollary III.12.3.
2737:{\displaystyle {\text{lct}}(X,f^{*}(p))}
2227:is some line bundle on the smooth curve
3273:
3233:
2198:{\displaystyle f^{*}(p_{i})=m_{i}D_{i}}
3393:
3313:
710:If the intersection matrix is affine A
699:If the intersection matrix is affine A
3344:
2454:associated to the singular fibers of
3355:Publications Mathématiques de l'IHÉS
3416:Mathematical classification systems
3009:minus the fiber over 0 by mapping (
2997:. We construct an isomorphism from
13:
1955:
1941:{\displaystyle {\tilde {\Gamma }}}
1929:
1458:
849:
342:v (v distinct intersection points)
304:2 (2 distinct intersection points)
148:Kodaira's table of singular fibers
24:is a surface that has an elliptic
14:
3427:
2245:holomorphic Euler characteristics
1988:, it is important to compute the
1911:For an elliptic fibration with a
739:) of 2 × 2 integer matrices with
692:), or have a double point (type I
2360:
1865:
1771:
1755:
1661:
1645:
1547:
1531:
1509:
1483:
1358:
1332:
1306:
1208:
1192:
1095:
1079:
985:
888:
871:
772:Group structure on smooth locus
666:
659:
632:
625:
598:
591:
564:
557:
526:
519:
489:
482:
455:
448:
439:2 (meet at one point of order 2)
421:
414:
358:
351:
320:
313:
283:
276:
249:
242:
40:such that almost all fibers are
3082:Enriques–Kodaira classification
2926:. Then the projection map from
2864:Logarithmic data transformation
2582:projective, the moduli divisor
2519:{\displaystyle (1/12)j^{*}O(1)}
169:Kodaira's symbol for the fiber,
3126:
3117:
3108:
3099:
2977:by this group action. We make
2804:{\displaystyle {}_{m}I_{\nu }}
2731:
2728:
2722:
2703:
2663:
2657:
2654:
2651:
2645:
2633:
2513:
2507:
2491:
2477:
2415:
2376:
2169:
2156:
2099:
2080:
2047:
2041:
1992:of a minimal elliptic surface
1932:
1496:
1479:
1345:
1328:
677:affine Dynkin diagram of type
1:
3319:Flips for 3-folds and 4-folds
3139:
2558:-linear equivalence class of
2941:There is an automorphism of
2836:{\displaystyle I_{\nu }^{*}}
2545:of the smooth fibers. (Thus
2135:Here the multiple fibers of
1872:{\displaystyle \mathbf {C} }
1313:{\displaystyle \mathbf {C} }
992:{\displaystyle \mathbf {C} }
722:
7:
3132:Kollár (2007), section 8.5.
3123:Kollár (2007), section 8.2.
3075:
2858:Logarithmic transformations
2538:is the function giving the
1917:Jacobian elliptic fibration
108:
10:
3432:
3001:minus the fiber over 0 to
2871:logarithmic transformation
2861:
2850:and others to families of
1986:classification of surfaces
3209:10.1007/978-1-4612-3696-2
3167:10.1007/978-3-642-57739-0
2446:is an explicit effective
2214:at least 2 and a divisor
3155:Compact Complex Surfaces
3092:
2981:into a fiber space over
2764:{\displaystyle I_{\nu }}
2139:(if any) are written as
2006:canonical bundle formula
1980:Canonical bundle formula
1894:potential good reduction
175:'s symbol for the fiber,
121:1 are elliptic surfaces.
3323:Oxford University Press
2690:log canonical threshold
2578:).) In particular, for
1961:{\displaystyle \Gamma }
1464:{\displaystyle \infty }
855:{\displaystyle \infty }
696:), or a cusp (type II).
473:3 (all meet in 1 point)
142:Shioda modular surfaces
79:and were classified by
2949:of order 2 that maps (
2918:be the elliptic curve
2837:
2805:
2765:
2738:
2670:
2520:
2425:
2199:
2126:
1962:
1942:
1873:
1847:
1779:
1737:
1669:
1627:
1555:
1517:
1465:
1444:
1366:
1314:
1290:
1216:
1174:
1103:
1061:
993:
967:
902:
856:
835:
2838:
2806:
2775:for a multiple fiber
2766:
2739:
2671:
2521:
2426:
2243:is determined by the
2200:
2127:
1963:
1943:
1874:
1848:
1780:
1738:
1670:
1628:
1556:
1518:
1466:
1445:
1367:
1315:
1291:
1217:
1175:
1104:
1062:
994:
968:
903:
857:
836:
270:1 (with double point)
3325:, pp. 134–162,
2852:Calabi–Yau varieties
2815:
2779:
2748:
2695:
2601:
2474:
2436:discriminant divisor
2341:
2334:-linear equivalence
2317:Building on work of
2143:
2015:
1952:
1923:
1861:
1801:
1751:
1691:
1641:
1578:
1527:
1476:
1455:
1392:
1325:
1302:
1241:
1188:
1125:
1075:
1015:
981:
921:
867:
846:
792:
191:affine Cartan matrix
3406:Birational geometry
3087:Néron minimal model
3048:Then the fibration
2969:be the quotient of
2832:
2564:divisor class group
760:Intersection matrix
216:Intersection matrix
52:, these fibers are
28:, in other words a
3411:Algebraic surfaces
3368:10.1007/BF02684271
3191:Cossec, François;
2854:of any dimension.
2833:
2818:
2801:
2761:
2734:
2666:
2632:
2516:
2421:
2195:
2122:
2079:
1974:Mordell-Weil group
1958:
1938:
1869:
1843:
1837:
1775:
1733:
1727:
1665:
1623:
1617:
1551:
1513:
1461:
1440:
1434:
1362:
1310:
1286:
1280:
1212:
1170:
1164:
1099:
1057:
1051:
989:
963:
957:
898:
852:
831:
825:
735:in the group SL(2,
389:with multiplicity
137:Dolgachev surfaces
62:proper base change
3275:Kodaira, Kunihiko
3235:Kodaira, Kunihiko
3197:Enriques Surfaces
3176:978-3-540-00832-3
2701:
2617:
2205:, for an integer
2070:
1935:
1882:
1881:
674:
673:
119:Kodaira dimension
69:complex manifolds
3423:
3401:Complex surfaces
3387:
3341:
3310:
3270:
3230:
3187:
3133:
3130:
3124:
3121:
3115:
3112:
3106:
3103:
2842:
2840:
2839:
2834:
2831:
2826:
2810:
2808:
2807:
2802:
2800:
2799:
2790:
2789:
2784:
2770:
2768:
2767:
2762:
2760:
2759:
2743:
2741:
2740:
2735:
2721:
2720:
2702:
2699:
2675:
2673:
2672:
2667:
2631:
2613:
2612:
2525:
2523:
2522:
2517:
2503:
2502:
2487:
2430:
2428:
2427:
2422:
2414:
2413:
2401:
2400:
2388:
2387:
2375:
2374:
2365:
2364:
2363:
2353:
2352:
2204:
2202:
2201:
2196:
2194:
2193:
2184:
2183:
2168:
2167:
2155:
2154:
2131:
2129:
2128:
2123:
2118:
2117:
2111:
2110:
2092:
2091:
2078:
2069:
2068:
2062:
2061:
2040:
2039:
2027:
2026:
1990:canonical bundle
1967:
1965:
1964:
1959:
1947:
1945:
1944:
1939:
1937:
1936:
1928:
1878:
1876:
1875:
1870:
1868:
1852:
1850:
1849:
1844:
1842:
1841:
1784:
1782:
1781:
1776:
1774:
1763:
1758:
1742:
1740:
1739:
1734:
1732:
1731:
1674:
1672:
1671:
1666:
1664:
1653:
1648:
1632:
1630:
1629:
1624:
1622:
1621:
1560:
1558:
1557:
1552:
1550:
1539:
1534:
1522:
1520:
1519:
1514:
1512:
1504:
1503:
1491:
1486:
1470:
1468:
1467:
1462:
1449:
1447:
1446:
1441:
1439:
1438:
1371:
1369:
1368:
1363:
1361:
1353:
1352:
1340:
1335:
1319:
1317:
1316:
1311:
1309:
1295:
1293:
1292:
1287:
1285:
1284:
1221:
1219:
1218:
1213:
1211:
1200:
1195:
1179:
1177:
1176:
1171:
1169:
1168:
1108:
1106:
1105:
1100:
1098:
1087:
1082:
1066:
1064:
1063:
1058:
1056:
1055:
998:
996:
995:
990:
988:
972:
970:
969:
964:
962:
961:
907:
905:
904:
899:
897:
896:
891:
879:
874:
861:
859:
858:
853:
840:
838:
837:
832:
830:
829:
754:
753:
670:
663:
636:
629:
602:
595:
568:
561:
530:
523:
493:
486:
459:
452:
425:
418:
362:
355:
324:
317:
287:
280:
253:
246:
204:
203:
155:Tate's algorithm
132:Kodaira surfaces
126:Enriques surface
117:All surfaces of
87:, especially in
81:Kunihiko Kodaira
22:elliptic surface
3431:
3430:
3426:
3425:
3424:
3422:
3421:
3420:
3391:
3390:
3291:10.2307/2373157
3251:10.2307/1970131
3219:
3193:Dolgachev, Igor
3177:
3142:
3137:
3136:
3131:
3127:
3122:
3118:
3113:
3109:
3104:
3100:
3095:
3078:
3072:with center 0.
2902:be the lattice
2867:
2860:
2848:Yujiro Kawamata
2827:
2822:
2816:
2813:
2812:
2795:
2791:
2785:
2783:
2782:
2780:
2777:
2776:
2755:
2751:
2749:
2746:
2745:
2716:
2712:
2698:
2696:
2693:
2692:
2621:
2608:
2604:
2602:
2599:
2598:
2590:
2553:
2498:
2494:
2483:
2475:
2472:
2471:
2469:
2445:
2409:
2405:
2396:
2392:
2383:
2379:
2370:
2366:
2359:
2358:
2354:
2348:
2344:
2342:
2339:
2338:
2329:
2293:
2284:
2271:
2222:
2213:
2189:
2185:
2179:
2175:
2163:
2159:
2150:
2146:
2144:
2141:
2140:
2113:
2112:
2106:
2102:
2087:
2083:
2074:
2064:
2063:
2057:
2053:
2035:
2031:
2022:
2018:
2016:
2013:
2012:
1982:
1953:
1950:
1949:
1927:
1926:
1924:
1921:
1920:
1887:
1864:
1862:
1859:
1858:
1836:
1835:
1830:
1824:
1823:
1815:
1805:
1804:
1802:
1799:
1798:
1795:
1770:
1759:
1754:
1752:
1749:
1748:
1726:
1725:
1720:
1714:
1713:
1705:
1695:
1694:
1692:
1689:
1688:
1685:
1660:
1649:
1644:
1642:
1639:
1638:
1616:
1615:
1610:
1604:
1603:
1595:
1582:
1581:
1579:
1576:
1575:
1572:
1546:
1535:
1530:
1528:
1525:
1524:
1508:
1499:
1495:
1487:
1482:
1477:
1474:
1473:
1456:
1453:
1452:
1433:
1432:
1424:
1418:
1417:
1409:
1396:
1395:
1393:
1390:
1389:
1386:
1379:
1357:
1348:
1344:
1336:
1331:
1326:
1323:
1322:
1305:
1303:
1300:
1299:
1279:
1278:
1270:
1264:
1263:
1258:
1245:
1244:
1242:
1239:
1238:
1235:
1229:
1207:
1196:
1191:
1189:
1186:
1185:
1163:
1162:
1154:
1145:
1144:
1139:
1129:
1128:
1126:
1123:
1122:
1119:
1094:
1083:
1078:
1076:
1073:
1072:
1050:
1049:
1044:
1035:
1034:
1029:
1019:
1018:
1016:
1013:
1012:
1009:
984:
982:
979:
978:
956:
955:
950:
941:
940:
935:
925:
924:
922:
919:
918:
892:
887:
886:
875:
870:
868:
865:
864:
847:
844:
843:
824:
823:
818:
812:
811:
806:
796:
795:
793:
790:
789:
786:
780:
733:conjugacy class
725:
717:
713:
706:
702:
695:
691:
656:
647:
622:
613:
588:
579:
554:
545:
538:
516:
507:
501:
479:
470:
445:
436:
405:
388:
375:
371:
348:
339:
332:
310:
301:
295:
267:
261:
230:
181:
150:
111:
77:singular fibers
73:regular schemes
54:elliptic curves
50:complex numbers
38:algebraic curve
32:with connected
30:proper morphism
12:
11:
5:
3429:
3419:
3418:
3413:
3408:
3403:
3389:
3388:
3342:
3311:
3271:
3231:
3217:
3188:
3175:
3147:Barth, Wolf P.
3141:
3138:
3135:
3134:
3125:
3116:
3107:
3097:
3096:
3094:
3091:
3090:
3089:
3084:
3077:
3074:
3060:. We say that
2859:
2856:
2830:
2825:
2821:
2798:
2794:
2788:
2771:, and it is 1/
2758:
2754:
2733:
2730:
2727:
2724:
2719:
2715:
2711:
2708:
2705:
2678:
2677:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2630:
2627:
2624:
2620:
2616:
2611:
2607:
2586:
2549:
2515:
2512:
2509:
2506:
2501:
2497:
2493:
2490:
2486:
2482:
2479:
2465:
2460:moduli divisor
2441:
2432:
2431:
2420:
2417:
2412:
2408:
2404:
2399:
2395:
2391:
2386:
2382:
2378:
2373:
2369:
2362:
2357:
2351:
2347:
2325:
2289:
2280:
2267:
2218:
2209:
2192:
2188:
2182:
2178:
2174:
2171:
2166:
2162:
2158:
2153:
2149:
2133:
2132:
2121:
2116:
2109:
2105:
2101:
2098:
2095:
2090:
2086:
2082:
2077:
2073:
2067:
2060:
2056:
2052:
2049:
2046:
2043:
2038:
2034:
2030:
2025:
2021:
1981:
1978:
1957:
1934:
1931:
1885:
1880:
1879:
1867:
1856:
1853:
1840:
1834:
1831:
1829:
1826:
1825:
1822:
1819:
1816:
1814:
1811:
1810:
1808:
1796:
1793:
1790:
1786:
1785:
1773:
1769:
1766:
1762:
1757:
1746:
1743:
1730:
1724:
1721:
1719:
1716:
1715:
1712:
1709:
1706:
1704:
1701:
1700:
1698:
1686:
1683:
1680:
1676:
1675:
1663:
1659:
1656:
1652:
1647:
1636:
1633:
1620:
1614:
1611:
1609:
1606:
1605:
1602:
1599:
1596:
1594:
1591:
1588:
1587:
1585:
1573:
1570:
1567:
1563:
1562:
1549:
1545:
1542:
1538:
1533:
1523:if ν is even,
1511:
1507:
1502:
1498:
1494:
1490:
1485:
1481:
1471:
1460:
1450:
1437:
1431:
1428:
1425:
1423:
1420:
1419:
1416:
1413:
1410:
1408:
1405:
1402:
1401:
1399:
1387:
1384:
1381:
1377:
1373:
1372:
1360:
1356:
1351:
1347:
1343:
1339:
1334:
1330:
1320:
1308:
1296:
1283:
1277:
1274:
1271:
1269:
1266:
1265:
1262:
1259:
1257:
1254:
1251:
1250:
1248:
1236:
1233:
1230:
1227:
1223:
1222:
1210:
1206:
1203:
1199:
1194:
1183:
1180:
1167:
1161:
1158:
1155:
1153:
1150:
1147:
1146:
1143:
1140:
1138:
1135:
1134:
1132:
1120:
1117:
1114:
1110:
1109:
1097:
1093:
1090:
1086:
1081:
1070:
1067:
1054:
1048:
1045:
1043:
1040:
1037:
1036:
1033:
1030:
1028:
1025:
1024:
1022:
1010:
1007:
1004:
1000:
999:
987:
976:
973:
960:
954:
951:
949:
946:
943:
942:
939:
936:
934:
931:
930:
928:
916:
913:
909:
908:
895:
890:
885:
882:
878:
873:
862:
851:
841:
828:
822:
819:
817:
814:
813:
810:
807:
805:
802:
801:
799:
787:
784:
781:
778:
774:
773:
770:
764:
761:
758:
724:
721:
720:
719:
715:
711:
708:
704:
700:
697:
693:
689:
672:
671:
664:
657:
654:
651:
648:
645:
642:
638:
637:
630:
623:
620:
617:
614:
611:
608:
604:
603:
596:
589:
586:
583:
580:
577:
574:
570:
569:
562:
555:
552:
549:
546:
543:
540:
536:
532:
531:
524:
517:
514:
511:
508:
505:
502:
499:
495:
494:
487:
480:
477:
474:
471:
468:
465:
461:
460:
453:
446:
443:
440:
437:
434:
431:
427:
426:
419:
412:
409:
406:
403:
400:
396:
395:
393:
386:
383:
381:
373:
367:
364:
363:
356:
349:
346:
343:
340:
337:
334:
330:
326:
325:
318:
311:
308:
305:
302:
299:
296:
293:
289:
288:
281:
274:
271:
268:
265:
262:
259:
255:
254:
247:
240:
237:
234:
231:
228:
224:
223:
220:
219:Dynkin diagram
217:
214:
211:
208:
202:
201:
198:
195:Dynkin diagram
183:
179:
176:
170:
149:
146:
145:
144:
139:
134:
129:
124:Every complex
122:
115:
110:
107:
9:
6:
4:
3:
2:
3428:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3398:
3396:
3385:
3381:
3377:
3373:
3369:
3365:
3361:
3358:(in French).
3357:
3356:
3351:
3347:
3343:
3340:
3336:
3332:
3328:
3324:
3320:
3316:
3315:Kollár, János
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3272:
3268:
3264:
3260:
3256:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3220:
3218:3-7643-3417-7
3214:
3210:
3206:
3202:
3198:
3194:
3189:
3186:
3182:
3178:
3172:
3168:
3164:
3160:
3156:
3152:
3148:
3144:
3143:
3129:
3120:
3111:
3102:
3098:
3088:
3085:
3083:
3080:
3079:
3073:
3071:
3067:
3063:
3059:
3055:
3051:
3046:
3044:
3040:
3036:
3032:
3028:
3024:
3020:
3016:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2960:
2956:
2952:
2948:
2944:
2939:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2890:
2888:
2884:
2880:
2876:
2872:
2865:
2855:
2853:
2849:
2844:
2828:
2823:
2819:
2796:
2792:
2786:
2774:
2756:
2752:
2725:
2717:
2713:
2709:
2706:
2691:
2687:
2683:
2660:
2648:
2642:
2639:
2636:
2628:
2625:
2622:
2618:
2614:
2609:
2605:
2597:
2596:
2595:
2592:
2589:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2552:
2548:
2544:
2542:
2537:
2533:
2529:
2510:
2504:
2499:
2495:
2488:
2484:
2480:
2468:
2464:
2461:
2457:
2453:
2449:
2444:
2440:
2437:
2418:
2410:
2406:
2402:
2397:
2393:
2389:
2384:
2380:
2371:
2367:
2355:
2349:
2345:
2337:
2336:
2335:
2333:
2328:
2324:
2320:
2315:
2313:
2309:
2305:
2301:
2297:
2292:
2288:
2283:
2279:
2275:
2270:
2266:
2262:
2258:
2254:
2250:
2246:
2242:
2238:
2234:
2230:
2226:
2221:
2217:
2212:
2208:
2190:
2186:
2180:
2176:
2172:
2164:
2160:
2151:
2147:
2138:
2119:
2107:
2103:
2096:
2093:
2088:
2084:
2075:
2071:
2058:
2054:
2050:
2044:
2036:
2032:
2028:
2023:
2019:
2011:
2010:
2009:
2007:
2003:
1999:
1995:
1991:
1987:
1977:
1975:
1971:
1918:
1914:
1909:
1907:
1903:
1899:
1895:
1891:
1857:
1854:
1838:
1832:
1827:
1820:
1817:
1812:
1806:
1797:
1791:
1788:
1787:
1767:
1764:
1760:
1747:
1744:
1728:
1722:
1717:
1710:
1707:
1702:
1696:
1687:
1681:
1678:
1677:
1657:
1654:
1650:
1637:
1634:
1618:
1612:
1607:
1600:
1597:
1592:
1589:
1583:
1574:
1568:
1565:
1564:
1543:
1540:
1536:
1505:
1500:
1492:
1488:
1472:
1451:
1435:
1429:
1426:
1421:
1414:
1411:
1406:
1403:
1397:
1388:
1382:
1375:
1374:
1354:
1349:
1341:
1337:
1321:
1297:
1281:
1275:
1272:
1267:
1260:
1255:
1252:
1246:
1237:
1231:
1225:
1224:
1204:
1201:
1197:
1184:
1181:
1165:
1159:
1156:
1151:
1148:
1141:
1136:
1130:
1121:
1115:
1112:
1111:
1091:
1088:
1084:
1071:
1068:
1052:
1046:
1041:
1038:
1031:
1026:
1020:
1011:
1005:
1002:
1001:
977:
974:
958:
952:
947:
944:
937:
932:
926:
917:
914:
911:
910:
893:
883:
880:
876:
863:
842:
826:
820:
815:
808:
803:
797:
788:
782:
776:
775:
771:
768:
765:
762:
759:
756:
755:
752:
750:
746:
742:
738:
734:
730:
709:
698:
687:
686:
685:
682:
680:
669:
665:
662:
658:
652:
649:
643:
640:
639:
635:
631:
628:
624:
618:
615:
609:
606:
605:
601:
597:
594:
590:
584:
581:
575:
572:
571:
567:
563:
560:
556:
550:
547:
541:
534:
533:
529:
525:
522:
518:
512:
509:
503:
497:
496:
492:
488:
485:
481:
475:
472:
466:
463:
462:
458:
454:
451:
447:
441:
438:
432:
429:
428:
424:
420:
417:
413:
410:
408:1 (with cusp)
407:
401:
398:
397:
394:
392:
384:
382:
379:
370:
366:
365:
361:
357:
354:
350:
344:
341:
335:
328:
327:
323:
319:
316:
312:
306:
303:
297:
291:
290:
286:
282:
279:
275:
272:
269:
263:
257:
256:
252:
248:
245:
241:
238:
235:
232:
226:
225:
221:
218:
215:
212:
209:
206:
205:
199:
196:
192:
188:
184:
177:
174:
171:
168:
167:
166:
163:
158:
156:
143:
140:
138:
135:
133:
130:
127:
123:
120:
116:
113:
112:
106:
104:
103:number fields
100:
97:
92:
90:
86:
85:string theory
82:
78:
74:
70:
65:
63:
59:
58:generic fiber
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
3359:
3353:
3346:Néron, André
3318:
3282:
3278:
3242:
3239:Ann. of Math
3238:
3196:
3154:
3151:Hulek, Klaus
3128:
3119:
3110:
3101:
3069:
3065:
3061:
3057:
3053:
3049:
3047:
3042:
3041:over all of
3038:
3034:
3030:
3026:
3022:
3018:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2986:
2985:by mapping (
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2954:
2950:
2946:
2942:
2940:
2935:
2931:
2927:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2894:
2891:
2886:
2882:
2878:
2877:with center
2874:
2870:
2868:
2845:
2772:
2689:
2685:
2681:
2679:
2593:
2587:
2583:
2579:
2575:
2572:Picard group
2567:
2559:
2555:
2550:
2546:
2540:
2535:
2531:
2527:
2466:
2462:
2459:
2455:
2451:
2450:-divisor on
2447:
2442:
2438:
2435:
2433:
2331:
2326:
2322:
2316:
2314:is minimal.
2311:
2307:
2303:
2302:-divisor on
2299:
2295:
2290:
2286:
2281:
2277:
2273:
2268:
2264:
2260:
2256:
2252:
2248:
2240:
2232:
2228:
2224:
2219:
2215:
2210:
2206:
2136:
2134:
2005:
2001:
1997:
1993:
1983:
1968:, as listed
1916:
1910:
1905:
1901:
1889:
1883:
1561:if ν is odd
766:
748:
736:
726:
683:
675:
390:
377:
368:
236:1 (elliptic)
159:
151:
93:
76:
66:
21:
15:
3285:: 751–798.
3279:Am. J. Math
3245:: 563–626.
1915:, called a
1898:j-invariant
741:determinant
187:zero matrix
173:André Néron
99:4-manifolds
18:mathematics
3395:Categories
3384:0132.41403
3307:0137.17501
3267:0118.15802
3201:Birkhäuser
3199:. Boston:
3140:References
2965:). We let
2914:, and let
2873:(of order
2811:, 1/2 for
2570:) and the
2543:-invariant
2458:, and the
2434:where the
2319:Kenji Ueno
769:-invariant
213:Components
44:curves of
3362:: 5–128.
2829:∗
2824:ν
2797:ν
2757:ν
2718:∗
2688:) is the
2640:−
2626:∈
2619:∑
2500:∗
2372:∗
2356:∼
2152:∗
2094:−
2072:∑
2051:⊗
2037:∗
1956:Γ
1933:~
1930:Γ
1818:−
1768:×
1708:−
1658:×
1598:−
1590:−
1544:×
1506:×
1459:∞
1427:−
1415:ν
1412:−
1404:−
1355:×
1273:−
1253:−
1205:×
1157:−
1149:−
1092:×
1039:−
945:−
894:∗
884:×
881:ν
850:∞
809:ν
763:Monodromy
729:monodromy
723:Monodromy
197:is given.
26:fibration
3348:(1964).
3195:(1989).
3159:Springer
3076:See also
2963:−s
2896:Example:
2526:, where
1792:affine E
1682:affine E
1569:affine E
1383:affine D
1232:affine D
1116:affine A
1006:affine A
783:affine A
745:homology
653:affine E
619:affine E
585:affine E
551:affine D
513:affine D
476:affine A
442:affine A
345:affine A
307:affine A
193:, whose
189:, or an
109:Examples
89:F-theory
3376:0179172
3339:2359346
3299:0187255
3259:0184257
3227:0986969
3185:2030225
2272:) − 2χ(
1913:section
207:Kodaira
162:minimal
3382:
3374:
3337:
3305:
3297:
3265:
3257:
3225:
3215:
3183:
3173:
3068:×
3056:×
3037:×
3025:)/2πi,
3017:) to (
3005:×
2973:×
2961:+1/2,
2957:) to (
2945:×
2930:×
2680:where
2259:) = χ(
2255:: deg(
2237:degree
376:(v≥0,
222:Fiber
96:smooth
42:smooth
36:to an
34:fibers
3093:Notes
3021:-log(
2993:) to
2554:is a
2231:. If
1380:(ν≥1)
757:Fiber
539:(v≥1)
333:(v≥2)
210:Néron
46:genus
20:, an
3213:ISBN
3171:ISBN
2898:Let
2574:Pic(
2251:and
1970:here
1745:1728
1069:1728
727:The
3380:Zbl
3364:doi
3327:doi
3303:Zbl
3287:doi
3263:Zbl
3247:doi
3205:doi
3163:doi
3045:.)
2934:to
2910:of
2700:lct
2566:Cl(
2470:is
2294:is
2247:of
2239:of
1679:III
1385:4+ν
1298:in
1003:III
785:ν-1
679:ADE
607:III
553:4+v
548:5+v
544:5,v
430:III
380:≥2)
347:v-1
71:or
16:In
3397::
3378:.
3372:MR
3370:.
3360:21
3352:.
3335:MR
3333:,
3321:,
3301:.
3295:MR
3293:.
3283:86
3281:.
3261:.
3255:MR
3253:.
3243:77
3241:.
3223:MR
3221:.
3211:.
3203:.
3181:MR
3179:,
3169:,
3161:,
3157:,
3149:;
2906:+i
2869:A
2534:→
2530::
2489:12
2310:→
2008::
2000:→
1996::
1789:II
1566:IV
1113:IV
912:II
681:.
641:II
573:IV
464:IV
399:II
105:.
91:.
64:.
3386:.
3366::
3329::
3309:.
3289::
3269:.
3249::
3229:.
3207::
3165::
3070:C
3066:E
3062:X
3058:C
3054:E
3050:X
3043:C
3039:C
3035:E
3031:X
3027:s
3023:s
3019:c
3015:s
3013:,
3011:c
3007:C
3003:E
2999:X
2995:s
2991:s
2989:,
2987:c
2983:C
2979:X
2975:C
2971:E
2967:X
2959:c
2955:s
2953:,
2951:c
2947:C
2943:E
2936:C
2932:C
2928:E
2924:L
2922:/
2920:C
2916:E
2912:C
2908:Z
2904:Z
2900:L
2887:m
2883:p
2879:p
2875:m
2866:.
2820:I
2793:I
2787:m
2773:m
2753:I
2732:)
2729:)
2726:p
2723:(
2714:f
2710:,
2707:X
2704:(
2686:p
2684:(
2682:c
2676:,
2664:]
2661:p
2658:[
2655:)
2652:)
2649:p
2646:(
2643:c
2637:1
2634:(
2629:S
2623:p
2615:=
2610:S
2606:B
2588:S
2584:M
2580:S
2576:S
2568:S
2560:Q
2556:Q
2551:S
2547:M
2541:j
2536:P
2532:S
2528:j
2514:)
2511:1
2508:(
2505:O
2496:j
2492:)
2485:/
2481:1
2478:(
2467:S
2463:M
2456:f
2452:S
2448:Q
2443:S
2439:B
2419:,
2416:)
2411:S
2407:M
2403:+
2398:S
2394:B
2390:+
2385:S
2381:K
2377:(
2368:f
2361:Q
2350:X
2346:K
2332:Q
2327:X
2323:K
2312:S
2308:X
2304:S
2300:Q
2296:Q
2291:X
2287:K
2282:S
2278:O
2276:,
2274:S
2269:X
2265:O
2263:,
2261:X
2257:L
2253:S
2249:X
2241:L
2233:S
2229:S
2225:L
2220:i
2216:D
2211:i
2207:m
2191:i
2187:D
2181:i
2177:m
2173:=
2170:)
2165:i
2161:p
2157:(
2148:f
2137:f
2120:.
2115:)
2108:i
2104:D
2100:)
2097:1
2089:i
2085:m
2081:(
2076:i
2066:(
2059:S
2055:O
2048:)
2045:L
2042:(
2033:f
2029:=
2024:X
2020:K
2002:S
1998:X
1994:f
1906:j
1902:j
1890:Z
1886:0
1866:C
1855:0
1839:)
1833:1
1828:1
1821:1
1813:0
1807:(
1794:8
1772:C
1765:2
1761:/
1756:Z
1729:)
1723:0
1718:1
1711:1
1703:0
1697:(
1684:7
1662:C
1655:3
1651:/
1646:Z
1635:0
1619:)
1613:0
1608:1
1601:1
1593:1
1584:(
1571:6
1548:C
1541:4
1537:/
1532:Z
1510:C
1501:2
1497:)
1493:2
1489:/
1484:Z
1480:(
1436:)
1430:1
1422:0
1407:1
1398:(
1378:ν
1376:I
1359:C
1350:2
1346:)
1342:2
1338:/
1333:Z
1329:(
1307:C
1282:)
1276:1
1268:0
1261:0
1256:1
1247:(
1234:4
1228:0
1226:I
1209:C
1202:3
1198:/
1193:Z
1182:0
1166:)
1160:1
1152:1
1142:1
1137:0
1131:(
1118:2
1096:C
1089:2
1085:/
1080:Z
1053:)
1047:0
1042:1
1032:1
1027:0
1021:(
1008:1
986:C
975:0
959:)
953:0
948:1
938:1
933:1
927:(
915:0
889:C
877:/
872:Z
827:)
821:1
816:0
804:1
798:(
779:ν
777:I
767:j
749:Z
737:Z
716:3
712:2
705:2
701:1
694:1
690:0
655:8
650:9
646:8
644:C
621:7
616:8
612:7
610:C
587:6
582:7
578:6
576:C
542:C
537:v
535:I
515:4
510:5
506:4
504:C
500:0
498:I
478:2
469:3
467:C
444:1
435:2
433:C
411:0
404:1
402:C
391:m
387:v
385:I
378:m
374:v
372:I
369:m
338:v
336:B
331:v
329:I
309:1
300:2
298:B
294:2
292:I
273:0
266:1
264:B
260:1
258:I
239:0
233:A
229:0
227:I
182:)
180:0
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.